More likely, one subset of humans (e.g. the ones who use birth control) will be replaced by another subset (e.g. the ones who don't use birth control, perhaps for religious reasons). Since evolution favors those who don't use birth control for obvious reasons.
The times when birth rate was our limiting factor are several centuries behind us. But good news, once our current population is decimated, we will also lack the means to produce viable birth control!
We're no longer limited to simple genetic reproduction, though... our units of cultural/technological/etc. output are also part of our legacies, whether as individuals or collectives. I think that's super cool, way more so than the old religious idea of "souls" and whatnot.
